Besides exact solutions, the long time asymptotic behavior of solutions of the Cauchy problem for nonlinear dispersive PDEs
has been studied extensively. The research on the long-time behavior
of nonlinear wave equations was firstly investigated with IST technique by Manakov in 1974 [1]}. By using the method,
Zakharov and Manakov presented the first result on the large time asymptotic behavior of solutions for the nonlinear Schrödinger equation
with decaying initial data [2]}. In a later development, Segur and Ablowitz extended this method
to derive the leading asymptotics for the solution of the mKdV, KdV and Sine-Gordon equations, including full
information on the phase [3]}. The most influential work to investigate the long-time behavior of integrable PDEs
is the nonlinear steepest descent method which was firstly proposed by Deift and Zhou (Deift-Zhou method) in 1993 [4]}.
In this work, Deift and Zhou developed rigorous analytic method to present the long-time asymptotic representation of the solution for defocusing mKdV equation
by deforming contours to reduce the original RH problem to a model whose solution can be derived in terms of parabolic cylinder functions for
Schwarz class initial data under zero boundary conditions. In 2016, Germain studied a full asymptotic stability for solitons of the
Cauchy problem for the focusing mKdV equation [5]}. And Griffiths made progress in the long-time asymptotics to the mKdV equation
in the same year [6]}. In [7]}, Lenells proved a nonlinear steepest descent theorem for RH problems
with Carleson jump contours and jump matrices of low regularity and slow decay. Recently, Chen and Liu extended the asymptotics to the solutions with initial data in lower regularity
spaces for focusing and defocusing mKdV equation [8]}, [9]}. For generic initial data in \(H^{1,1}(\mathbb {R})\) , Wang and Ma studied the asymptotics
for defocusing mKdV equation under the zero boundary conditions [10]}.
